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Molecular Dynamics Simulations of Noble Gas Release from Endohedral Fullerene Clusters

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Molecular Dynamics Simulations of Noble Gas Release from Endohedral Fullerene Clusters Molecular dynamics simulations of noble gas release from endohedral fullerene clusters M.K. BALASUBRAMANYA*†, M.W. ROTH‡, P.D. TILTON† and B.A. SUCHY‡ † Department of Physical and Life Sciences, Texas A&M University-Corpus Christi, Corpus Christi, Texas 78412, USA. ‡ Department of Physics, University of Northern Iowa, Cedar Falls, Iowa 50614, USA. We report the results of molecular dynamics simulations of the release of five species of noble gas atoms trapped inside a small cluster of fullerenes in the temperature range 4000K ≤ T ≤ 5000K. We find that larger noble gas atoms are generally released at a slower rate and that helium is released considerably more rapidly than any of the other noble gases. The differing release rates are due not only to the differences in the size and mass of a given endohedral species but also because larger trapped atoms tend to stabilize the fullerene cage against thermal fluctuations. Unlike with the case of atoms entering fullerenes, we find that any atom escaping from the cage results in a window which does not close. Escape rate constants are reported and comparisons with experiment are discussed. Keywords: Fullerene cluster; Endofullerene; Noble gas; Molecular dynamics; Simulation. 1. Introduction Endohedral fullerenes, or carbon cages trapping atomic or molecular species, have received significant attention both experimentally [1-16] and theoretically [17-33]. Such systems with noble gas atoms trapped inside the molecular cage are formed while making fullerenes by passing an electric arc between carbon electrodes in an inert atmosphere of noble gases. Much interest has focused on the behavior of these systems for primarily two reasons. Endofullerenes are found terrestrially at meteor sites with 3He trapped inside. Their study can throw light on their extraterrestrial origins, especially the prevalent conditions at the time of their formation [34]. Secondly, chemists have been interested in encapsulating noble gas atoms inside fullerene cages and study the interactions between the host and guest. Cross and Saunders have pioneered the insertion 3 of He into C60 [35]. This endohedral molecule is chemically modified outside the cage in different ways and subjected to NMR analysis. Since every 3He-labeled fullerene has a distinctive helium chemical shift, that shift can be used to pin down the structure of the derivative, as well as monitor the molecule's subsequent chemical transformations. 3He NMR spectroscopy has thus become one of the most powerful tools for following fullerene chemistry. In addition to He, four other noble gases - Ne, Ar, Kr and Xe - have been inserted into fullerenes, making unusual and highly stable noble gas compounds in which no formal bond exists between the noble gas and the surrounding carbon atoms. A very convenient way to experimentally probe an endohedral fullerene system is to raise its temperature until the encapsulated species is released, and to subsequently measure the concentration of the released species. Measurements have been made of the release of Ne from endohedral Ne@C60 [15]. It is possible for the fullerene to release a Ne atom without the fullerene structure being destroyed, which is impossible if the Ne atom is simply pushed through the molecular cage, breaking the C-C bonds. Moreover, in the presence of impurities, the rate of release of trapped noble gas atoms is increased by 1 Figure 1. Initial conditions utilized for the simulations. The five fullerenes in the cluster form one face of an FCC fullerite unit cell with lattice constants a = b = 14.4 Å. The carbon atoms in each fullerene are colored green if they are closest to its center of mass, red if they are farthest away and a mixture of red and green if they are in between. The orange atoms inside the fullerene cages are encapsulated Ne atoms, and the relative atomic sizes, chosen for visual clarity, are not to scale. 2 orders of magnitude. A modified windowing mechanism has therefore been proposed, where the impurity (e.g. radical) adds to the cage and weakens fullerene bonds. The endohedral atom, according to this model, exits from the ‘weak spot’ of the cage, or its ‘window’, followed by the impurity detaching from the carbon atoms cluster, thus allowing reconstitution of the C-C bonds and the fullerene cage [15]. We conducted molecular dynamics (MD) simulations of the release of Ne from small Ne@C60 clusters without impurities [33], and found that the simulations describe the system reasonably well as far as overall cluster dynamics and individual fullerene disintegration is concerned, but not when dealing with windowing at temperatures as low as seen experimentally. We strongly suspect that a modification of the character of the bonds in the MD simulations would be required to adequately describe the windowing suspected in real systems, but even then direct modeling of this process will require computational times of the order of the presently accepted age of the universe. Much remains to be understood regarding the process of release of endohedral species from fullerene systems. Despite their limitations, MD simulations have provided a reasonable description of, and considerable insight into, fullerene systems [17-33]. Moreover, the noble gas atoms are a family of chemically similar species that differ mainly in their size and mass and, as such, they serve as ideal candidates for behaviour comparison in endofullerenes. The purpose of this study is to enhance our understanding of experimental and simulated endohedral release of noble gas atoms from fullerene systems by a comparative MD computer simulation. This study focuses on the release of five noble gas atoms encapsulated in C60 clusters. 2. Computational Approach The Ne@C60 cluster chosen for this study has five endohedral fullerenes. The cluster size is chosen to be small because the process of release takes a substantial amount of simulated time. With a smaller cluster size it is possible to do many runs and obtain reasonable statistics. Moreover, in a real cluster containing many more fullerenes, as the temperature rises, smaller crystallites leave the cluster edge and it is likely that endohedral release happens in the gas phase from such small free crystallites. For this reason, periodic boundary conditions are not utilized; we wish to simulate small clusters where edge effects are important and cluster dissociation is not stifled. There is a very large reflecting box the cluster is kept in so that the system volume is constant. However none of the particles ever reflect off this wall in our simulation, so in actuality we implement free boundary conditions on the cluster. Above 257 K the fullerite crystal forms an FCC lattice. We model the initial configuration of the cluster at every temperature as one face of an FCC unit cell which has sheared off from the cell. As simulated time runs forward, the equations of motion are integrated using a standard Verlet algorithm with a time step ∆t=0.0005 ps, and various structural averages, thermodynamic averages and relative atomic position distributions are calculated. In the temperature range 4000K ≤ T ≤ 5000K, the results of 5 different runs are averaged at temperatures spaced 50 K apart, and temperature control is achieved by velocity rescaling for the carbon atoms and the noble gas population separately. Based on endohedral 3 release times and the degree of equilibration of the system, each run is taken out to 2x106 time steps, or 1 ns. The initial configuration for the simulations is shown in Figure 1. 350 300 He 250 200 150 100 50 0 0 2 4 6 8 10 12 14 16 350 300 Ne 250 200 150 100 50 0 0 2 4 6 8 10 12 14 16 350 300 Xe 250 200 150 100 50 0 0 2 4 6 8 10 12 14 16 Figure 2. Fullerene pair distribution function Pf(rij) at T = 4000 K (blue), T = 4500K (green) and T = 5000K (purple) for He, Ne and Xe. The horizontal axes are in Angstrom and the vertical axes are arbitrary units; all axes are to the same scale. 4 There are several types of interaction potentials used in the simulations. The noble gas–noble gas potential as well as the noble gas-carbon potential are of a Lennard-Jones form, » 12 6 ÿ ≈σ ’ ≈σ ’ u (r ) = 4ε …∆ ij ÷ − ∆ ij ÷ Ÿ, (1) LJ ij ij …∆ r ÷ ∆ r ÷ Ÿ « ij ◊ « ij ◊ ⁄ where the potential parameters for interaction between various species are given in table 1. Mixed interaction parameters are obtained with the use of Lorentz-Bertholot combining rules involving carbon-carbon parameters for the same potential as in equation 1. In addition, there is a non-bonded carbon-carbon interaction which is in a modified Lennard-Jones form [29], » 12 6 ≈σ ’ ≈σ ’ u~ (r ) = ε …∆ CC ÷ − 2∆ CC ÷ (2) LJ ij CC …∆ r ÷ ∆ r ÷ « ij ◊ « ij ◊ whose parameters are also shown in table 1. Table 1. Parameters for the non-bonded Lennard-Jones (LJ) interaction potentials. The interactions with asterisks (*) are not used explicitly in the simulations because they are for a standard LJ interaction, not the modified one actually used in this study. They are used only in the combining rule relationships to get noble gas-carbon interaction parameters in the LJ potential. Species εij(K) σij (Å) He-He 10.80 2.57 Ne-Ne 36.68 2.79 Ar-Ar 120.0 3.38 Kr-Kr 171.0 3.60 Xe-Xe 221.0 4.10 C-C* 28.00* 3.40* C-C 34.839 3.805 The non-bonded carbon–carbon potential parameters given in table 1, and used in equation 2, are not derivable from the potential and the parameters in equation 1. The asterisked parameters (for the traditional Lennard–Jones interaction) apply to fullerene adsorption onto graphite [36] while the parameters for the modified Lennard–Jones potential for atomic carbon–carbon interactions apply to non-bonded fullerene carbons [29].
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